Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations

In this paper we study Hyers-Ulam stability of Euler’s equation in the calculus of variations in two special cases: when F=F(x,y′) and when F=F(y,y′). For the first case we use the direct method and for the second case we use the Laplace transform. In the first Theorem and in the first Example the corresponding estimations for Jyx−Jy0x are given. We mention that it is the first time that the problem of Ulam-stability of extremals for functionals represented in integral form is studied.

A cohomological obstruction in higher-dimensional Chern–Simons gauge theories

We study a set of cohomology classes which emerge in the cohomological formulations of the calculus of variations as obstructions to the existence of (global) solutions of the Euler–Lagrange equations of Chern–Simons gauge theories in higher dimensions [Formula: see text].

Morse theory and the calculus of variations

We prove for the first time that classical Morse theory applies to functionals of the form 𝒥 ⁢ ( u ) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ ( x ) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 𝑑 x + ∫ Ω G ⁢ ( x , u ) ⁢ 𝑑 x \displaystyle\mathcal{J}(u)=\frac{1}{2}\int_{\Omega}A^{ij}_{\alpha\beta}(x)% \frac{\partial u^{i}}{\partial x^{\alpha}}\frac{\partial u^{j}}{\partial x^{% \beta}}\,dx+\int_{\Omega}G(x,u)\,dx where u : Ω → ℝ N {u:\Omega\to\mathbb{R}^{N}} , Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} compact with C ∞ {C^{\infty}} boundary ∂ ⁡ Ω {\partial\Omega} , u | ∂ ⁡ Ω = φ {u|_{\partial\Omega}=\varphi} , and we argue that this is the largest class to which Morse theory applies.

Best cylindrical map projections according to the undesirability of angular and areal distortions

Seeking low distortion maps, it is usual to assume that areal and angular distortions are equally undesirable on the map. However, this might not be the case for certain map thematics. Should angular distortions be a bit less preferred to areal distortions, maps of unbalanced distortions may be developed. In this paper, the known analytic solution for the best cylindrical map projection is extended to such more general requirements by utilizing calculus of variations. The overall distortion of the resulted mappings are calculated and compared to each other to explore the distortion characteristics of these intentionally unbalanced map projections.